CBSE Class 12 Sample Paper for 2024 Boards

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Question 17 - CBSE Class 12 Sample Paper for 2024 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Last updated at Feb. 10, 2025 by

The set of all points where the function f(x)=x+|x| is differentiable, is

(a) (0,โˆž)ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย  (b) (-โˆž,0)

(c) (-โˆž,0)โˆช(0,โˆž)ย  ย  ย  ย  ย  ย  ย  ย  ย  ย  ย  ย  (d) (-โˆž,โˆž)

This question is similar to Question-15 NCERT-Exemplar-MCQs

The set of all points where function f(x) = x + |x| is differentiable - CBSE Class 12 Sample Paper for 2024 Boards

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Transcript

๐‘“(๐‘ฅ) = ๐‘ฅ+|๐‘ฅ| = {โ–ˆ(๐‘ฅ+๐‘ฅ, ๐‘ฅโ‰ฅ0@๐‘ฅโˆ’๐‘ฅ, ๐‘ฅ <0)โ”ค = {โ–ˆ(2๐‘ฅ, ๐‘ฅโ‰ฅ0@0, ๐‘ฅ<0)โ”ค Now, f(x) is a differentiable at x = 0 if LHD = RHD (๐’๐’Š๐’Ž)โ”ฌ(๐กโ†’๐ŸŽ) (๐’‡(๐’™) โˆ’ ๐’‡(๐’™ โˆ’ ๐’‰))/๐’‰ = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘“(0) โˆ’ ๐‘“(0 โˆ’ โ„Ž))/โ„Ž = (๐’๐’Š๐’Ž)โ”ฌ(๐กโ†’๐ŸŽ) (๐’‡(๐ŸŽ) โˆ’ ๐’‡(โˆ’๐’‰))/๐’‰ =(๐‘™๐‘–๐‘š)โ”ฌ(โ„Žโ†’0) (2(0) โˆ’ 0)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) 0/โ„Ž = ๐ŸŽ (๐’๐’Š๐’Ž)โ”ฌ(๐กโ†’๐ŸŽ) (๐’‡(๐’™ + ๐’‰) โˆ’ ๐’‡(๐’™ ))/๐’‰ = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘“(0 + โ„Ž) โˆ’ ๐‘“(0))/โ„Ž = (๐’๐’Š๐’Ž)โ”ฌ(๐กโ†’๐ŸŽ) (๐’‡(๐’‰) โˆ’ ๐’‡(๐ŸŽ))/๐’‰ = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (2โ„Ž โˆ’ 0)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) 2โ„Ž/โ„Ž = ๐Ÿ Hence, we can say that f(x) is differentiable on R โˆ’ {๐ŸŽ} or (โˆ’โˆž,๐ŸŽ)โˆช(๐ŸŽ,โˆž) So, the correct answer is (c)

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 15 years. He provides courses for Maths, Science and Computer Science at Teachoo